Distribution of final obliquities and semimajor axes of HJs in our three-planet simulations. Panel (a) corresponds to our fiducial setup. In panels (b)–(d), we include an additional outer planet with mass 8 M_Jup. The location of the outermost planet is set by the parameter ﹩{ \mathcal R }﹩ (see text for details); it lies farthest out for ﹩{ \mathcal R }=0.01﹩ and closest in for ﹩{ \mathcal R }=0.4﹩. Panels (a) and (b) are similar, indicating that a distant outer planet does not significantly alter the final distribution of semimajor axes and obliquities. For ﹩{ \mathcal R }=0.1﹩ and ﹩{ \mathcal R }=0.4﹩, all planets in the system are tightly coupled. Even in these cases, we recover the envelope relating obliquity to semimajor axis, demonstrating the robustness of our fiducial results.